We propose and analyze an efficient algorithm for solving the joint sparse
recovery problem using a new regularization-based method, named orthogonally
weighted $\ell_{2,1}$ ($\mathit{ow}\ell_{2,1}$), which is specifically designed
to take into account the rank of the solution matrix. This method has
applications in feature extraction, matrix column selection, and dictionary
learning, and it is distinct from commonly used $\ell_{2,1}$ regularization and
other existing regularization-based approaches because it can exploit the full
rank of the row-sparse solution matrix, a key feature in many applications. We
provide a proof of the method's rank-awareness, establish the existence of
solutions to the proposed optimization problem, and develop an efficient
algorithm for solving it, whose convergence is analyzed. We also present
numerical experiments to illustrate the theory and demonstrate the
effectiveness of our method on real-life problems.

我们提出并分析了一种高效的算法，用于解决联合稀疏恢复问题，采用一种名为正交加权 ℓ2,1 （owℓ2,1）的新的正则化方法，专门设计用于考虑解矩阵的秩。该方法在特征提取、矩阵列选择和词典学习等方面有应用，与常用的 ℓ2,1 正则化和其他现有的正则化方法不同，因为它可以利用行稀疏解矩阵的完全秩，这是许多应用中的关键特性。我们提供了该方法秩感知性质的证明，建立了所提优化问题的解的存在性，并开发了一个高效的算法来求解该问题，并对其收敛性进行了分析。我们还展示了数值实验，以说明理论并展示我们的方法在实际问题上的有效性。